Question: Solve for $t$, $ \dfrac{3t - 9}{5t - 15} = -\dfrac{6}{t - 3} + \dfrac{7}{t - 3} $
First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5t - 15$ $t - 3$ and $t - 3$ The common denominator is $5t - 15$ The denominator of the first term is already $5t - 15$ , so we don't need to change it. To get $5t - 15$ in the denominator of the second term, multiply it by $\frac{5}{5}$ $ -\dfrac{6}{t - 3} \times \dfrac{5}{5} = -\dfrac{30}{5t - 15} $ To get $5t - 15$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ \dfrac{7}{t - 3} \times \dfrac{5}{5} = \dfrac{35}{5t - 15} $ This give us: $ \dfrac{3t - 9}{5t - 15} = -\dfrac{30}{5t - 15} + \dfrac{35}{5t - 15} $ If we multiply both sides of the equation by $5t - 15$ , we get: $ 3t - 9 = -30 + 35$ $ 3t - 9 = 5$ $ 3t = 14 $ $ t = \dfrac{14}{3}$